On an Arithmetic Convolution
نویسندگان
چکیده
منابع مشابه
On an Arithmetic Convolution
The Cauchy-type product of two arithmetic functions f and g on nonnegative integers is defined by (f • g)(k) := ∑k m=0 ( k m ) f(m)g(k −m). We explore some algebraic properties of the aforementioned convolution, which is a fundamental characteristic of the identities involving the Bernoulli numbers, the Bernoulli polynomials, the power sums, the sums of products, and so forth.
متن کاملThe convolution inverse of an arithmetic function
Good, but does an inverse g of f have to exist? A necessary condition is that f(1) 6= 0. Indeed, if g is the inverse of f , then 1 = I(1) = (f ∗ g)(1) = f(1)g(1). We now show that this necessary condition is also sufficient. We assume that f(1) 6= 0 and we try and solve for g. What this means is that we are solving for infinitely many unknowns: g(1), g(2), . . . . From the above, we see that th...
متن کاملArithmetic Convolution Rings
Arithmetic convolution rings provide a general and unified treatment of many rings that have been called arithmetic; the best known examples are rings of complex valued functions with domain in the set of non-negative integers and multiplication the Cauchy product or the Dirichlet product. The emphasis here is on factorization and related properties of such rings which necessitates prior result...
متن کاملConvolution Structures and Arithmetic Cohomology
In the beginning of 1998 Gerard van de Geer and René Schoof posted a beautiful preprint (cf. [2]). Among other things in this preprint they defined exactly h(L) for Arakelov line bundles L on an “arithmetic curve”, i.e. a number field. The main advantage of their definition was that they got an exact analog of the Riemann-Roch formula h(L) − h0(K−L) = degL+1−g. Before that h(L) was defined as a...
متن کاملConvolution Structures and Arithmetic Cohomology
In the beginning of 1998 Gerard van der Geer and Ren e Schoof posted a beautiful preprint (cf. [2]). Among other things in this preprint they de ned exactly h(L) for Arakelov line bundles L on an \arithmetic curve", i.e. a number eld. The main advantage of their de nition was that they got an exact analog of the Riemann-Roch formula h(L) h(K L) = degL+1 g: Before that h(L) was de ned as an inte...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Canadian Mathematical Bulletin
سال: 1977
ISSN: 0008-4395,1496-4287
DOI: 10.4153/cmb-1977-046-9